The stability of independence polynomials of complete bipartite graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909628491628544 |
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| author | Chen, Guo Ning, Bo Tu, Jianhua |
| author_facet | Chen, Guo Ning, Bo Tu, Jianhua |
| contents | The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z \in \mathbb{C} : \mathrm{Re}(z) \leq 0\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \textbf{Are all complete bipartite graphs stable?}
We answer this question by establishing the following results:
\begin{itemize}
\item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable.
\item For any integer $k\geq0$, there exists an integer $N(k)\in \mathbb{N}$ such that $K_{m,m+k}$ is stable for all $m>N(k)$.
\item For any rational $\ell> 1$, there exists an integer $N(\ell) \in \mathbb{N}$ such that whenever $m >N(\ell)$ and $\ell \cdot m$ is an integer, $K_{m, \ell \cdot m}$ is \textbf{not} stable.
\end{itemize} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The stability of independence polynomials of complete bipartite graphs Chen, Guo Ning, Bo Tu, Jianhua Combinatorics The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z \in \mathbb{C} : \mathrm{Re}(z) \leq 0\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \textbf{Are all complete bipartite graphs stable?} We answer this question by establishing the following results: \begin{itemize} \item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable. \item For any integer $k\geq0$, there exists an integer $N(k)\in \mathbb{N}$ such that $K_{m,m+k}$ is stable for all $m>N(k)$. \item For any rational $\ell> 1$, there exists an integer $N(\ell) \in \mathbb{N}$ such that whenever $m >N(\ell)$ and $\ell \cdot m$ is an integer, $K_{m, \ell \cdot m}$ is \textbf{not} stable. \end{itemize} |
| title | The stability of independence polynomials of complete bipartite graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.24381 |